Journal of Global Optimization

, Volume 64, Issue 2, pp 305–323 | Cite as

On refinement of the unit simplex using regular simplices

  • B. G.-Tóth
  • E. M. T. Hendrix
  • L. G. Casado
  • I. García


A natural way to define branching in branch and bound (B&B) for blending problems is bisection. The consequence of using bisection is that partition sets are in general irregular. The question is how to use regular simplices in the refinement of the unit simplex. A regular simplex with fixed orientation can be represented by its center and size, facilitating storage of the search tree from a computational perspective. The problem is that a simplex defined in a space with dimension \(n>3\) cannot be subdivided into regular subsimplices without overlapping. We study the characteristics of the refinement by regular simplices. The main challenge is to find a refinement with a good convergence ratio which allows discarding simplices in an overlapped and already evaluated region. As the efficiency of the division rule in B&B algorithms is instance dependent, we focus on the worst case behaviour, i.e. none of the branches are pruned. This paper shows that for this case surprisingly an overlapping regular refinement may generate less simplices to be evaluated than longest edge bisection. On the other hand, the number of evaluated vertices may be larger.


Unit simplex Branch and bound Partition Covering 


  1. 1.
    Aparicio, G., Casado, L.G., Tóth, B.G., Hendrix, E.M.T., García, I.: On the minimim number of simplex shapes in longest edge bisection refinement of a regular \(n\)-simplex. Informatica 26(1), 17–32 (2015)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Casado, L.G., García, I., Tóth, B., Hendrix, E.M.T.: On determining the cover of a simplex by spheres centered at its vertices. J. Glob. Optim. 50(4), 645–655 (2010)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Casado, L.G., Hendrix, E.M.T., García, I.: Infeasibility spheres for finding robust solutions of blending problems with quadratic constraints. J. Glob. Optim. 39(2), 215–236 (2007)MathSciNetMATHGoogle Scholar
  4. 4.
    Claussen, J., Žilinskas, A.: Subdivision, sampling and initialization strategies for simplical branch and bound in global optimization. Comput. Math. Appl. 44, 943–955 (2002)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    de Klerk, E.: The complexity of optimizing over a simplex, hypercube or sphere: a short survey. CEJOR 16(2), 111–125 (2008)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Dickinson, P.J.C.: On the exhaustivity of simplicial partitioning. J. Glob. Optim. 58(1), 189–203 (2014)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Hannukainen, A., Korotov, S., Křížek, M.: On numerical regularity of the face-to-face longest-edge bisection algorithm for tetrahedral partitions. Science of Computer Programming, 90, Part A(0):34 – 41, (2014). Special issue on Numerical Software: design, analysis and verificationGoogle Scholar
  8. 8.
    Hendrix, E.M.T., Casado, L.G., Amaral, P.: Global optimization simplex bisection revisited based on considerations by Reiner Horst. In: Computational Science and Its Applications ICCSA 2012, volume 7335 of Lecture Notes in Computer Science, pp. 159–173. Springer, Berlin, (2012)Google Scholar
  9. 9.
    Hendrix, E.M.T., Casado, L.G., García, I.: The semi-continuous quadratic mixture design problem: description and branch-and-bound approach. Eur. J. Oper. Res. 191(3), 803–815 (2008)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Horst, R.: On generalized bisection of \(n\)-simplices. Math. Comput. 66(218), 691–698 (1997)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Horst, R.: Bisecton by global optimization revisited. J. Optim. Theory Appl. 144(3), 501–510 (2010)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Korotov, S., Kek, M.: Red refinements of simplices into congruent subsimplices. Comput. Math. Appl. 67(12), 2199–2204 (2014)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Žilinskas, J., Paulavičius, R.: Simplicial Global Optimization. Springer, Berlin (2014)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • B. G.-Tóth
    • 1
  • E. M. T. Hendrix
    • 2
  • L. G. Casado
    • 3
  • I. García
    • 4
  1. 1.Dept. of Differential EquationsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Operations Research and LogisticsWageningen UniversityWageningenThe Netherlands
  3. 3.Dept. of Informatics, Agrifood Campus of International Excellence (ceiA3)Universidad de AlmeríaAlmeríaSpain
  4. 4.Computer ArchitectureUniversidad de MálagaMálagaSpain

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